Integrated risk management process

ABSTRACT

A method and system allowing the analysis of risk through the use of Monte Carlo simulation, statistical and data analysis, stochastic forecasting, and optimization. The present invention includes novel methods such as the detailed reporting capabilities coupled with advanced analytical techniques, an integrated risk management process and procedures, adaptive licensing technology, and model profiling and storage procedures.

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BACKGROUND OF THE INVENTION

The present invention is in the field of finance, economics, math, andbusiness statistics, and relates to the modeling and valuation of riskwithin all companies, allowing these firms to properly assess, quantify,value, diversify, and hedge their risks.

The field of risk analysis is large and complex, and this inventionallows a new and novel set of analytics in a integrated andcomprehensive manner, where executive reports with detailedexplanations, numerical results and charts are generated within one tothree mouse clicks. The invention uses Monte Carlo simulation,stochastic forecasting, business statistics, and optimization techniquesin a novel way to analyze a user's existing data set to extract valuableand important information. In addition, the invention includes new andnovel computer logic to save modeling parameters and inputs withinmultiple profiles in a single Excel workbook file, and covers a uniquemethod to license software products through the use of a hardwareidentification algorithm.

SUMMARY OF THE INVENTION

Risk and uncertainty abound in the business world and impact businessdecisions and ultimately affects the profitability and survival of thecorporation. The present invention's preferred embodiment isencapsulated in the Risk Simulator software, which incorporates a lot ofadvanced analytical techniques and algorithms and compiles them in sucha unique and novel way to facilitate business risk analysis, through anintelligent set of statistical and analytical tests to analyze andextract information that otherwise cannot be obtained manually. That is,instead of requiring the user to understand advanced statistics,financial modeling and mathematics, in order to know what analysis torun on some existing data or the ability to interpret the raw numericalresults, this present invention automatically runs the relevant analysesin an integrated fashion, and provides detailed description in itsreports, coupled with the numerical results and charts for easyinterpretation. The present invention also includes a novel licensingcapability that extracts the user's system and hardware information tocreate a license protection. In addition, an integrated risk managementbusiness process method is developed that allows the user to stepthrough the risk analysis methodology step by step in an integrated andcomprehensive manner.

Monte Carlo simulation refers to a method where risk and uncertainty isquantified, through the use of mathematical algorithms of randomlysampling numbers from a specific distribution. For instance, suppose weneed to forecast the revenues of a product the following year but thisvalue is unknown, but nonetheless, we know from past experience thatrevenues for these types of products has a mean of X and standarddeviation of Y, and follows a normal distribution. Further suppose thatthere are multiple products sold by this company. We can then takeadvantage of this fact and randomly select data points thousands oftimes with replacement, from a set of normal distribution with thesespecifications. The end result is thousands of forecast results, andusing these results, we can determine the company's total revenues andthe probability that the revenue will exceed some predefined value, andso forth.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 01 illustrates the Integrated Risk Management Process steps.

FIG. 02 illustrates the Risk Simulator menu item with integratedlanguage capabilities.

FIG. 03 illustrates the licensing scheme using a computer's unique setof hardware information.

FIG. 04 illustrates the simulation profiles.

FIG. 05 illustrates a sample report from the integrated system.

FIG. 06 illustrates the econometrics modeling capabilities in thesystem.

FIG. 07 illustrates the stochastic forecasting module and report.

FIG. 08 illustrates the auto ARIMA module and report.

FIG. 09 illustrates the distribution analysis tool and associatedprobabilities.

FIG. 10 illustrates the integrated statistical data analysis module.

FIG. 11 illustrates the statistical data analysis report structure.

FIG. 12 illustrates the econometric and regression data diagnosticmodule.

FIG. 13 illustrates the econometric and regression diagnostics reportingstructure.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 01 illustrates the integrated risk management process 1 on theprocess and method undertaken in the preferred embodiment of the presentinvention in the Risk Simulator software.

FIG. 02 illustrates the menu items in the software and that severalforeign languages 2 exists in the software and can be changedimmediately without having to restart the operating system or having toinstall a different software version. All language packs are availableimmediately.

FIG. 03 illustrates the licensing schema. The present invention's methodallows the software to access the user computer's hardware and softwareconfigurations such as the user name on the computer, serial number onthe operating system, serial numbers from various hardware devices suchas the hard drive, motherboard, wireless and Ethernet card, take thesevalues and apply some proprietary mathematical algorithms to convertthem into a 10 to 20 alphanumerical Hardware ID 3. These Hardware IDsare unique to each computer and no two computers have the sameidentification. The prefix to this Hardware ID indicates the softwaretype while the last letter on the ID indicates the type of hardwareconfiguration on this computer (e.g., the letter “F” indicates that thehard drive and motherboard are properly installed and these serialnumbers are used to generate this ID). Other suffix letters indicatevarious combinations of serial numbers used.

FIG. 04 illustrates the profile method in the software 4, where insteadof creating multiple versions of the same Excel file, a single Excelfile can contain multiple profiles that stores all the relevantinformation and parameters in the model, the system comprising theability to create, duplicate, delete, and switch among multiple profilesin a single Excel file and the profile is saved as an encrypted XML codein a hidden sheet in the Excel file.

FIG. 05 illustrates a sample report generated from the software,complete with numerical results 5, detailed descriptions of the results6 and executive dashboards and charts 7.

FIG. 06 illustrates the basic econometrics method in the software, wheredata is loaded directly into the tool 8 and customizable functions andequations can be entered 9 and a sample set of results 10 are availableprior to running the entire report.

FIG. 07 illustrates another sample report 11 and method calledstochastic process forecasting 12, where multiple types of stochasticmodels are available to forecast future values (e.g., Brownian motionrandom walk for forecasting stock prices and commodity asset prices,mean-reversion for forecasting interest rates and inflation rates,jump-diffusion for modeling utility, oil and gas prices).

FIG. 08 illustrates a novel method called Auto-ARIMA 13 where differentcombinations of ARIMA (autoregressive integrated moving average) models(see the section on the mathematical details of this modeling approach)are automatically run and analyzed in this report, returning therankings of the best to the worst forecasting models.

FIG. 09 illustrates the distributional analysis tool 14 with 24statistical distributions 15 where the probability density function(PDF), cumulative distribution function (CDF) and inverse cumulativedistribution function (ICDF) 17 are available. The results show thedistributional chart and probability tables 18.

FIG. 10 illustrates a comprehensive and integrated statistical analysismethod where given some data, a set of comprehensive data analysistechniques are available 19 within a single mouse click.

FIG. 11 illustrates the results generated using the statistical analysistool method, where we obtain detailed descriptive statistics 20,distributional fitting 21, hypothesis testing results 22, nonlinearextrapolation 23, normality test 24, stochastic parameter estimations25, autocorrelation 26, time-series autocorrelation 27, and linear trend28.

FIG. 12 illustrates a comprehensive and integrated forecasting datadiagnostic method where given some data, a set of comprehensive dataanalysis techniques are available 29 within a single mouse click.

FIG. 13 illustrates the results of the data diagnostic tool, where thetest for heteroskedasticity, micronumerosity, outliers and nonlinearity30 are tested, followed by tests on normality and sphericity of theerrors 31, autocorrelation 32, stochastic parameter estimations 33,multicollinearity 34, and correlation significance analysis 35.

Mathematical Probability Distributions

This section demonstrates the mathematical models and computations usedin creating the Monte Carlo simulations. In order to get started withsimulation, one first needs to understand the concept of probabilitydistributions. To begin to understand probability, consider thisexample: You want to look at the distribution of nonexempt wages withinone department of a large company. First, you gather raw data—in thiscase, the wages of each nonexempt employee in the department. Second,you organize the data into a meaningful format and plot the data as afrequency distribution on a chart. To create a frequency distribution,you divide the wages into group intervals and list these intervals onthe chart's horizontal axis. Then you list the number or frequency ofemployees in each interval on the chart's vertical axis. Now you caneasily see the distribution of nonexempt wages within the department.You can chart this data as a probability distribution. A probabilitydistribution shows the number of employees in each interval as afraction of the total number of employees. To create a probabilitydistribution, you divide the number of employees in each interval by thetotal number of employees and list the results on the chart's verticalaxis.

Probability distributions are either discrete or continuous. Discreteprobability distributions describe distinct values, usually integers,with no intermediate values and are shown as a series of vertical bars.A discrete distribution, for example, might describe the number of headsin four flips of a coin as 0, 1, 2, 3, or 4. Continuous probabilitydistributions are actually mathematical abstractions because they assumethe existence of every possible intermediate value between two numbers;that is, a continuous distribution assumes there is an infinite numberof values between any two points in the distribution. However, in manysituations, you can effectively use a continuous distribution toapproximate a discrete distribution even though the continuous modeldoes not necessarily describe the situation exactly.

Probability Density Functions, Cumulative Distribution Functions, andProbability Mass Functions

In mathematics and Monte Carlo simulation, a probability densityfunction (PDF) represents a continuous probability distribution in termsof integrals. If a probability distribution has a density of ƒ(x), thenintuitively the infinitesimal interval of [x, x+dx] has a probability ofƒ(x) dx. The PDF therefore can be seen as a smoothed version of aprobability histogram; that is, by providing an empirically large sampleof a continuous random variable repeatedly, the histogram using verynarrow ranges will resemble the random variable's PDF. The probabilityof the interval between [a, b] is given by

∫_(a)^(b)f(x)x,

which means that the total integral of the function ƒ must be 1.0. It isa common mistake to think of ƒ(a) as the probability of a. This isincorrect. In fact, ƒ(a) can sometimes be larger than 1—consider auniform distribution between 0.0 and 0.5. The random variable x withinthis distribution will have ƒ(x) greater than 1. The probability inreality is the function ƒ(x)dx discussed previously, where dx is aninfinitesimal amount.

The cumulative distribution function (CDF) is denoted as F(x)=P(X≦x)indicating the probability of X taking on a less than or equal value tox. Every CDF is monotonically increasing, is continuous from the right,and at the limits, have the following properties:

${\lim\limits_{x{- \infty}}{F(x)}} = {{0\mspace{14mu} {and}\mspace{14mu} {\lim\limits_{x{+ \infty}}{F(x)}}} = 1.}$

Further, the CDF is related to the PDF by

F(b) − F(a) = P(a ≤ X ≤ b) = ∫_(a)^(b)f(x)x,

where the PDF function ƒ is the derivative of the CDF function F.

In probability theory, a probability mass function or PMF gives theprobability that a discrete random variable is exactly equal to somevalue. The PMF differs from the PDF in that the values of the latter,defined only for continuous random variables, are not probabilities;rather, its integral over a set of possible values of the randomvariable is a probability. A random variable is discrete if itsprobability distribution is discrete and can be characterized by a PMF.Therefore, X is a discrete random variable if

${\sum\limits_{u}{P\left( {X = u} \right)}} = 1$

as u runs through all possible values of the random variable X.

Discrete Distributions

Following is a detailed listing of the different types of probabilitydistributions that can be used in Monte Carlo simulation.

Bernoulli or Yes/No Distribution

The Bernoulli distribution is a discrete distribution with two outcomes(e.g., head or tails, success or failure, 0 or 1). The Bernoullidistribution is the binomial distribution with one trial and can be usedto simulate Yes/No or Success/Failure conditions. This distribution isthe fundamental building block of other more complex distributions. Forinstance:

-   -   Binomial distribution: Bernoulli distribution with higher number        of n total trials and computes the probability of x successes        within this total number of trials.    -   Geometric distribution: Bernoulli distribution with higher        number of trials and computes the number of failures required        before the first success occurs.    -   Negative binomial distribution: Bernoulli distribution with        higher number of trials and computes the number of failures        before the xth success occurs.

The mathematical constructs for the Bernoulli distribution are asfollows:

${P(x)} = \left\{ {{\begin{matrix}{1 - p} & {{{for}\mspace{14mu} x} = 0} \\p & {{{for}\mspace{14mu} x} = 1}\end{matrix}{or}{P(x)}} = {{{p^{x}\left( {1 - p} \right)}^{1 - x}{mean}} = {{p{standard}\mspace{14mu} {deviation}} = {{\sqrt{p\left( {1 - p} \right)}{skewness}} = {{\frac{1 - {2p}}{\sqrt{p\left( {1 - p} \right)}}{excess}\mspace{14mu} {kurtosis}} = \frac{{6p^{2}} - {6p} + 1}{p\left( {1 - p} \right)}}}}}} \right.$

The probability of success (p) is the only distributional parameter.Also, it is important to note that there is only one trial in theBernoulli distribution, and the resulting simulated value is either 0or 1. The input requirements are such thatProbability of Success>0 and <1 (that is, 0.0001≦p≦0.9999).

Binomial Distribution

The binomial distribution describes the number of times a particularevent occurs in a fixed number of trials, such as the number of heads in10 flips of a coin or the number of defective items out of 50 itemschosen.

The three conditions underlying the binomial distribution are:

-   -   For each trial, only two outcomes are possible that are mutually        exclusive.    -   The trials are independent-what happens in the first trial does        not affect the next trial.    -   The probability of an event occurring remains the same from        trial to trial.

The mathematical constructs for the binomial distribution are asfollows:

${P(x)} = {\frac{n!}{{x!}{\left( {n - x} \right)!}}{p^{x}\left( {1 - p} \right)}^{({n - x})}}$for  n > 0; x = 0, 1, 2, …  n; and  0 < p < 1 mean = np${{standard}\mspace{14mu} {deviation}} = \sqrt{{np}\left( {1 - p} \right)}$${skewness} = \frac{1 - {2p}}{\sqrt{{np}\left( {1 - p} \right)}}$${{excess}\mspace{14mu} {kurtosis}} = \frac{{6p^{2}} - {6p} + 1}{{np}\left( {1 - p} \right)}$

The probability of success (p) and the integer number of total trials(n) are the distributional parameters. The number of successful trialsis denoted x. It is important to note that probability of success (p) of0 or 1 are trivial conditions and do not require any simulations, andhence, are not allowed in the software. The input requirements are suchthat Probability of Success>0 and <1 (that is, 0.0001≦p≦0.9999), theNumber of Trials>1 or positive integers and ≦1000 (for larger trials,use the normal distribution with the relevant computed binomial mean andstandard deviation as the normal distribution's parameters).

Discrete Uniform

The discrete uniform distribution is also known as the equally likelyoutcomes distribution, where the distribution has a set of N elements,and each element has the same probability. This distribution is relatedto the uniform distribution but its elements are discrete and notcontinuous. The mathematical constructs for the discrete uniformdistribution are as follows:

${P(x)} = \frac{1}{N}$${mean} = {\frac{N + 1}{2}\mspace{14mu} {ranked}\mspace{14mu} {value}}$${{standard}\mspace{14mu} {deviation}} = {\sqrt{\frac{\left( {N - 1} \right)\left( {N + 1} \right)}{12}}\mspace{14mu} {ranked}\mspace{14mu} {value}}$skewness = 0  (that  is, the  distribution  is  perfectly  symmetrical)${{excess}\mspace{14mu} {kurtosis}} = {\frac{{- 6}\left( {N^{2} + 1} \right)}{5\left( {N - 1} \right)\left( {N + 1} \right)}\mspace{14mu} {ranked}\mspace{14mu} {value}}$

The input requirements are such that Minimum<Maximum and both must beintegers (negative integers and zero are allowed).

Geometric Distribution

The geometric distribution describes the number of trials until thefirst successful occurrence, such as the number of times you need tospin a roulette wheel before you win.

The three conditions underlying the geometric distribution are:

-   -   The number of trials is not fixed.    -   The trials continue until the first success.    -   The probability of success is the same from trial to trial.

The mathematical constructs for the geometric distribution are asfollows:

P(x) = p(1 − p)^(x − 1)  for  0 < p < 1  and  x = 1, 2, …  , n${mean} = {\frac{1}{p} - 1}$${{standard}\mspace{14mu} {deviation}} = \sqrt{\frac{1 - p}{p^{2}}}$${skewness} = \frac{2 - p}{\sqrt{1 - p}}$${{excess}\mspace{14mu} {kurtosis}} = \frac{p^{2} - {6p} + 6}{1 - p}$

The probability of success (p) is the only distributional parameter. Thenumber of successful trials simulated is denoted x, which can only takeon positive integers. The input requirements are such that Probabilityof success>0 and <1 (that is, 0.0001≦p≦0.9999). It is important to notethat probability of success (p) of 0 or 1 are trivial conditions and donot require any simulations, and hence, are not allowed in the software.

Hypergeometric Distribution

The hypergeometric distribution is similar to the binomial distributionin that both describe the number of times a particular event occurs in afixed number of trials. The difference is that binomial distributiontrials are independent, whereas hypergeometric distribution trialschange the probability for each subsequent trial and are called trialswithout replacement. For example, suppose a box of manufactured parts isknown to contain some defective parts. You choose a part from the box,find it is defective, and remove the part from the box. If you chooseanother part from the box, the probability that it is defective issomewhat lower than for the first part because you have removed adefective part. If you had replaced the defective part, theprobabilities would have remained the same, and the process would havesatisfied the conditions for a binomial distribution.

The three conditions underlying the hypergeometric distribution are:

-   -   The total number of items or elements (the population size) is a        fixed number, a finite population. The population size must be        less than or equal to 1,750.    -   The sample size (the number of trials) represents a portion of        the population.    -   The known initial probability of success in the population        changes after each trial.

The mathematical constructs for the hypergeometric distribution are asfollows:

$\mspace{79mu} {{P(x)} = \frac{\frac{\left( N_{x} \right)!}{{x!}{\left( {N_{x} - x} \right)!}}\frac{\left( {N - N_{x}} \right)!}{{\left( {n - x} \right)!}{\left( {N - N_{x} - n + x} \right)!}}}{\frac{N!}{{n!}{\left( {N - n} \right)!}}}}$     for  x = Max(n − (N − N_(x)), 0), …  , Min(n, N_(x))$\mspace{79mu} {{mean} = \frac{N_{x}n}{N}}$$\mspace{79mu} {{{standard}\mspace{14mu} {deviation}} = \sqrt{\frac{\left( {N - N_{x}} \right)N_{x}{n\left( {N - n} \right)}}{N^{2}\left( {N - 1} \right)}}}$$\mspace{79mu} {{skewness} = {\frac{\left( {N - {2N_{x}}} \right)\left( {N - {2n}} \right)}{N - 2}\sqrt{\frac{N - 1}{\left( {N - N_{x}} \right)N_{x}{n\left( {N - n} \right)}}}}}$$\mspace{79mu} {{{excess}\mspace{14mu} {kurtosis}} = {\frac{V\left( {N,N_{x},n} \right)}{\left( {N - N_{x}} \right)N_{x}{n\left( {{- 3} + N} \right)}\left( {{- 2} + N} \right)\left( {{- N} + n} \right)}\mspace{14mu} {where}}}$V(N, N_(x), n) = (N − N_(x))³ − (N − N_(x))⁵ + 3(N − N_(x))²N_(x) − 6(N − N_(x))³N_(x) + (N − N_(x))⁴N_(x) + 3(N − N_(x))N_(x)² − 12(N − N_(x))²N_(x)² + 8(N − N_(x))³N_(x)² + N_(x)³ − 6(N − N_(x))N_(x)³ + 8(N − N_(x))²N_(x)³ + (N − N_(x))N_(x)⁴ − N_(x)⁵ − 6(N − N_(x))³N_(x) + 6(N − N_(x))⁴N_(x) + 18(N − N_(x))²N_(x)n − 6(N − N_(x))³N_(x)n + 18(N − N_(x))N_(x)²n − 24(N − N_(x))²N_(x)²n − 6(N − N_(x))³n − 6(N − N_(x))N_(x)³n + 6N_(x)⁴n + 6(N − N_(x))²n² − 6(N − N_(x))³n² − 24(N − N_(x))N_(x)n² + 12(N − N_(x))²N_(x)n² + 6N_(x)²n² + 12(N − N_(x))N_(x)²n² − 6N_(x)³n²

The number of items in the population (N), trials sampled (n), andnumber of items in the population that have the successful trait (N_(x))are the distributional parameters. The number of successful trials isdenoted x. The input requirements are such that Population≧2 andinteger,Trials>0 and integerSuccesses>0 and integer, Population>Successes

Trials<Population and Population<1750. Negative Binomial Distribution

The negative binomial distribution is useful for modeling thedistribution of the number of trials until the rth successfuloccurrence, such as the number of sales calls you need to make to closea total of 10 orders. It is essentially a superdistribution of thegeometric distribution. This distribution shows the probabilities ofeach number of trials in excess of r to produce the required success r.

Conditions

The three conditions underlying the negative binomial distribution are:

-   -   The number of trials is not fixed.    -   The trials continue until the rth success.    -   The probability of success is the same from trial to trial.

The mathematical constructs for the negative binomial distribution areas follows:

${P(x)} = {\frac{\left( {x + r - 1} \right)!}{{\left( {r - 1} \right)!}{x!}}{p^{r}\left( {1 - p} \right)}^{x}}$for  x = r, r + 1, …  ; and  0 < p < 1${mean} = \frac{r\left( {1 - p} \right)}{p}$${{standard}\mspace{14mu} {deviation}} = \sqrt{\frac{r\left( {1 - p} \right)}{p^{2}}}$${skewness} = \frac{2 - p}{\sqrt{r\left( {1 - p} \right)}}$${{excess}\mspace{14mu} {k{urtosis}}} = \frac{p^{2} - {6p} + 6}{r\left( {1 - p} \right)}$

Probability of success (p) and required successes (r) are thedistributional parameters. Where the input requirements are such thatSuccesses required must be positive integers>0 and <8000, Probability ofsuccess>0 and <1 (that is, 0.0001≦p≦0.9999). It is important to notethat probability of success (p) of 0 or 1 are trivial conditions and donot require any simulations, and hence, are not allowed in the software.

Poisson Distribution

The Poisson distribution describes the number of times an event occursin a given interval, such as the number of telephone calls per minute orthe number of errors per page in a document.

Conditions

The three conditions underlying the Poisson distribution are:

-   -   The number of possible occurrences in any interval is unlimited.    -   The occurrences are independent. The number of occurrences in        one interval does not affect the number of occurrences in other        intervals.    -   The average number of occurrences must remain the same from        interval to interval.

The mathematical constructs for the Poisson are as follows:

${P(x)} = {{\frac{^{- \lambda}\lambda^{x}}{x!}\mspace{14mu} {for}\mspace{14mu} x\mspace{14mu} {and}\mspace{14mu} \lambda} > 0}$mean = λ ${{standard}\mspace{14mu} {deviation}} = \sqrt{\lambda}$${skewness} = \frac{1}{\sqrt{\lambda}}$${{excess}\mspace{14mu} {kurtosis}} = \frac{1}{\lambda}$

Rate (λ) is the only distributional parameter and the input requirementsare such that Rate>0 and ≦1000 (that is, 0.0001≦rate≦1000).

Continuous Distributions Beta Distribution

The beta distribution is very flexible and is commonly used to representvariability over a fixed range. One of the more important applicationsof the beta distribution is its use as a conjugate distribution for theparameter of a Bernoulli distribution. In this application, the betadistribution is used to represent the uncertainty in the probability ofoccurrence of an event. It is also used to describe empirical data andpredict the random behavior of percentages and fractions, as the rangeof outcomes is typically between 0 and 1. The value of the betadistribution lies in the wide variety of shapes it can assume when youvary the two parameters, alpha and beta. If the parameters are equal,the distribution is symmetrical. If either parameter is 1 and the otherparameter is greater than 1, the distribution is J-shaped. If alpha isless than beta, the distribution is said to be positively skewed (mostof the values are near the minimum value). If alpha is greater thanbeta, the distribution is negatively skewed (most of the values are nearthe maximum value). The mathematical constructs for the betadistribution are as follows:

${{f(x)} = {{\frac{(x)^{({\alpha - 1})}\left( {1 - x} \right)^{({\beta - 1})}}{\left\lbrack \frac{{\Gamma (\alpha)}{\Gamma (\beta)}}{\Gamma \left( {\alpha + \beta} \right)} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} \alpha} > 0}};{\beta > 0};{x > 0}$${mean} = \frac{\alpha}{\alpha + \beta}$${{standard}\mspace{14mu} {deviation}} = \sqrt{\frac{\alpha\beta}{\left( {\alpha + \beta} \right)^{2}\left( {1 + \alpha + \beta} \right)}}$${skewness} = \frac{2\left( {\beta - \alpha} \right)\sqrt{1 + \alpha + \beta}}{\left( {2 + \alpha + \beta} \right)\sqrt{\alpha\beta}}$${{excess}\mspace{14mu} {kurtosis}} = {\frac{3{\left( {\alpha + \beta + 1} \right)\left\lbrack {{{\alpha\beta}\left( {\alpha + \beta - 6} \right)} + {2\left( {\alpha + \beta} \right)^{2}}} \right\rbrack}}{{{\alpha\beta}\left( {\alpha + \beta + 2} \right)}\left( {\alpha + \beta + 3} \right)} - 3}$

Alpha (α) and beta (β) are the two distributional shape parameters, andΓ is the gamma function.

The two conditions underlying the beta distribution are:

-   -   The uncertain variable is a random value between 0 and a        positive value.    -   The shape of the distribution can be specified using two        positive values.

Input requirements:

Alpha and beta>0 and can be any positive value

Cauchy Distribution or Lorentzian Distribution or Breit-WignerDistribution

The Cauchy distribution, also called the Lorentzian distribution orBreit-Wigner distribution, is a continuous distribution describingresonance behavior. It also describes the distribution of horizontaldistances at which a line segment tilted at a random angle cuts thex-axis.

The mathematical constructs for the cauchy or Lorentzian distributionare as follows:

${f(x)} = {\frac{1}{\pi}\frac{\gamma/2}{\left( {x - m} \right)^{2} + {\gamma^{2}/4}}}$

The cauchy distribution is a special case where it does not have anytheoretical moments (mean, standard deviation, skewness, and kurtosis)as they are all undefined. Mode location (m) and scale (γ) are the onlytwo parameters in this distribution. The location parameter specifiesthe peak or mode of the distribution while the scale parameter specifiesthe half-width at half-maximum of the distribution. In addition, themean and variance of a cauchy or Lorentzian distribution are undefined.In addition, the cauchy distribution is the Student's t distributionwith only 1 degree of freedom. This distribution is also constructed bytaking the ratio of two standard normal distributions (normaldistributions with a mean of zero and a variance of one) that areindependent of one another. The input requirements are such thatLocation can be any value whereas Scale>0 and can be any positive value.

Chi-Square Distribution

The chi-square distribution is a probability distribution usedpredominately in hypothesis testing, and is related to the gammadistribution and the standard normal distribution. For instance, thesums of independent normal distributions are distributed as a chi-square(χ²) with k degrees of freedom:

Z₁ ²+Z₂ ²+ . . . +Z_(k) ² ^(d) ˜χ_(k) ²

The mathematical constructs for the chi-square distribution are asfollows:

${f(x)} = {{\frac{2^{{- k}/2}}{\Gamma \left( {k/2} \right)}x^{{k/2} - 1}^{{- x}/2}\mspace{14mu} {for}\mspace{14mu} {all}\mspace{14mu} x} > 0}$mean = k ${{standard}\mspace{14mu} {deviation}} = \sqrt{2k}$${skewness} = {2\sqrt{\frac{2}{k}}}$${{excess}\mspace{14mu} {kurtosis}} = \frac{12}{k}$

Γ is the gamma function. Degrees of freedom k is the only distributionalparameter.

The chi-square distribution can also be modeled using a gammadistribution by setting the

${{shape}\mspace{14mu} {parameter}} = \frac{k}{2}$

and scale=2S² where S is the scale. The input requirements are such thatDegrees of freedom>1 and must be an integer<1000.

Exponential Distribution

The exponential distribution is widely used to describe events recurringat random points in time, such as the time between failures ofelectronic equipment or the time between arrivals at a service booth. Itis related to the Poisson distribution, which describes the number ofoccurrences of an event in a given interval of time. An importantcharacteristic of the exponential distribution is the “memoryless”property, which means that the future lifetime of a given object has thesame distribution, regardless of the time it existed. In other words,time has no effect on future outcomes. The mathematical constructs forthe exponential distribution are as follows:

f(x) = λ ^(−λ x)  for  x ≥ 0; λ > 0${mean} = \frac{1}{\lambda}$${{standard}\mspace{14mu} {deviation}} = \frac{1}{\lambda}$skewness = 2  (this  value  applies  to  all  success  rate  λ  inputs)excess  kurtosis = 6  (this  value  applies  to  all  success  rate  λ  inputs)

Success rate (λ) is the only distributional parameter. The number ofsuccessful trials is denoted x.

The condition underlying the exponential distribution is:

-   -   The exponential distribution describes the amount of time        between occurrences.        Input requirements: Rate>0 and ≦300

Extreme Value Distribution or Gumbel Distribution

The extreme value distribution (Type 1) is commonly used to describe thelargest value of a response over a period of time, for example, in floodflows, rainfall, and earthquakes. Other applications include thebreaking strengths of materials, construction design, and aircraft loadsand tolerances. The extreme value distribution is also known as theGumbel distribution.

The mathematical constructs for the extreme value distribution are asfollows:

$\mspace{79mu} {{f(x)} = {{\frac{1}{\beta}z\; ^{- Z}\mspace{14mu} {where}\mspace{14mu} z} = ^{\frac{x - m}{\beta}}}}$     for  β > 0; and  any  value  of  x  and  m     mean = m + 0.577215β$\mspace{79mu} {{{standard}\mspace{14mu} {deviation}} = \sqrt{\frac{1}{6}\pi^{2}\beta^{2}}}$${skewness} = {\frac{12\sqrt{6}(1.2020569)}{\pi^{3}} = {1.13955\mspace{14mu} \left( {{this}\mspace{14mu} {applies}\mspace{14mu} {for}\mspace{14mu} {all}\mspace{14mu} {values}\mspace{14mu} {of}\mspace{14mu} {mode}\mspace{14mu} {and}\mspace{14mu} {scale}} \right)}}$excess  kurtosis = 5.4  (this  applies  for  all  values  of  mode  and  scale)

Mode (m) and scale (β) are the distributional parameters. There are twostandard parameters for the extreme value distribution: mode and scale.The mode parameter is the most likely value for the variable (thehighest point on the probability distribution). The scale parameter is anumber greater than 0. The larger the scale parameter, the greater thevariance. The input requirements are such that Mode can be any value andScale>0.

F Distribution or Fisher-Snedecor Distribution

The F distribution, also known as the Fisher-Snedecor distribution, isanother continuous distribution used most frequently for hypothesistesting. Specifically, it is used to test the statistical differencebetween two variances in analysis of variance tests and likelihood ratiotests. The F distribution with the numerator degree of freedom n anddenominator degree of freedom m is related to the chi-squaredistribution in that:

$\left. \frac{\chi_{n}^{2}/n^{d}}{\chi_{m}^{2}/m} \right.\sim F_{n,m}$${{or}\mspace{14mu} {f(x)}} = \frac{{\Gamma \left( \frac{n + m}{2} \right)}\left( \frac{n}{m} \right)^{n/2}x^{{n/2} - 1}}{{\Gamma \left( \frac{n}{2} \right)}{{\Gamma \left( \frac{m}{2} \right)}\left\lbrack {{x\left( \frac{n}{m} \right)} + 1} \right\rbrack}^{{({n + m})}/2}}$${mean} = \frac{m}{m - 2}$${{standard}\mspace{14mu} {deviation}} = {{\frac{2{m^{2}\left( {m + n - 2} \right)}}{{n\left( {m - 2} \right)}^{2}\left( {m - 4} \right)}\mspace{14mu} {for}\mspace{14mu} {all}\mspace{14mu} m} > 4}$${skewness} = {\frac{2\left( {m + {2n} - 2} \right)}{m - 6}\sqrt{\frac{2\left( {m - 4} \right)}{n\left( {m + n - 2} \right)}}}$${{excess}\mspace{14mu} {kurtosis}} = \frac{\begin{matrix}{12\left( {{- 16} + {20m} - {8m^{2}} + m^{3} +} \right.} \\{{44n} - {32{mn}} + {5m^{2}n} - {22n^{2}} + {5{mn}^{2}}}\end{matrix}}{{n\left( {m - 6} \right)}\left( {m - 8} \right)\left( {n + m - 2} \right)}$

The numerator degree of freedom n and denominator degree of freedom mare the only distributional parameters. The input requirements are suchthat Degrees of freedom numerator and degrees of freedom denominatorboth>0 integers.

Gamma Distribution (Erlang Distribution)

The gamma distribution applies to a wide range of physical quantitiesand is related to other distributions: lognormal, exponential, Pascal,Erlang, Poisson, and Chi-Square. It is used in meteorological processesto represent pollutant concentrations and precipitation quantities. Thegamma distribution is also used to measure the time between theoccurrence of events when the event process is not completely random.Other applications of the gamma distribution include inventory control,economic theory, and insurance risk theory.

The gamma distribution is most often used as the distribution of theamount of time until the rth occurrence of an event in a Poissonprocess. When used in this fashion, the three conditions underlying thegamma distribution are:

-   -   The number of possible occurrences in any unit of measurement is        not limited to a fixed number.    -   The occurrences are independent. The number of occurrences in        one unit of measurement does not affect the number of        occurrences in other units.    -   The average number of occurrences must remain the same from unit        to unit.

The mathematical constructs for the gamma distribution are as follows:

${f(x)} = {{\frac{\left( \frac{x}{\beta} \right)^{\alpha - 1}^{- \frac{x}{\beta}}}{{\Gamma (\alpha)}\beta}\mspace{14mu} {with}\mspace{14mu} {any}\mspace{14mu} {value}\mspace{14mu} {of}\mspace{14mu} \alpha} < {0\mspace{14mu} {and}\mspace{14mu} \beta} > 0}$mean = α β${{standard}\mspace{14mu} {deviation}} = \sqrt{\alpha \; \beta^{2}}$${skewness} = \frac{2}{\sqrt{\alpha}}$${{{excess}\mspace{14mu} {kurtosis}} = \frac{6}{\alpha}}\mspace{14mu}$

Shape parameter alpha (α) and scale parameter beta (β) are thedistributional parameters, and Γ is the gamma function. When the alphaparameter is a positive integer, the gamma distribution is called theErlang distribution, used to predict waiting times in queuing systems,where the Erlang distribution is the sum of independent and identicallydistributed random variables each having a memoryless exponentialdistribution. Setting n as the number of these random variables, themathematical construct of the Erlang distribution is:

${f(x)} = {{\frac{x^{n - 1}^{- x}}{\left( {n - 1} \right)!}\mspace{14mu} {for}\mspace{14mu} {all}\mspace{14mu} x} > 0}$

and all positive integers of n, where the input requirements are suchthat Scale Beta>0 and can be any positive value, Shape Alpha≧0.05 andany positive value, and Location can be any value.

Logistic Distribution

The logistic distribution is commonly used to describe growth, that is,the size of a population expressed as a function of a time variable. Italso can be used to describe chemical reactions and the course of growthfor a population or individual.

The mathematical constructs for the logistic distribution are asfollows:

${f(x)} = {\frac{^{\frac{\mu - x}{\alpha}}}{{\alpha \left\lbrack {1 + ^{\frac{\mu - x}{\alpha}}} \right\rbrack}^{2}}\mspace{14mu} {for}\mspace{14mu} {any}\mspace{14mu} {value}\mspace{14mu} {of}\mspace{14mu} \alpha \mspace{14mu} {and}\mspace{14mu} \mu}$mean = μ${{standard}\mspace{14mu} {deviation}} = \sqrt{\frac{1}{3}\pi^{2}\alpha^{2}}$skewness = 0(this  applies  to  all  mean  and  scale  inputs)excess  kurtosis = 1.2(this  applies  to  all  mean  and  scale  inputs)

Mean (μ) and scale (α) are the distributional parameters. There are twostandard parameters for the logistic distribution: mean and scale. Themean parameter is the average value, which for this distribution is thesame as the mode, because this distribution is symmetrical. The scaleparameter is a number greater than 0. The larger the scale parameter,the greater the variance.

Input requirements:

Scale>0 and can be any positive valueMean can be any value

Lognormal Distribution

The lognormal distribution is widely used in situations where values arepositively skewed, for example, in financial analysis for securityvaluation or in real estate for property valuation, and where valuescannot fall below zero. Stock prices are usually positively skewedrather than normally (symmetrically) distributed. Stock prices exhibitthis trend because they cannot fall below the lower limit of zero butmight increase to any price without limit. Similarly, real estate pricesillustrate positive skewness and are lognormally distributed as propertyvalues cannot become negative.

The three conditions underlying the lognormal distribution are:

-   -   The uncertain variable can increase without limits but cannot        fall below zero.    -   The uncertain variable is positively skewed, with most of the        values near the lower limit.    -   The natural logarithm of the uncertain variable yields a normal        distribution.

Generally, if the coefficient of variability is greater than 30 percent,use a lognormal distribution. Otherwise, use the normal distribution.

The mathematical constructs for the lognormal distribution are asfollows:

${f(x)} = {\frac{1}{x\sqrt{2\pi}{\ln (\sigma)}}^{- \frac{{\lbrack{{\ln {(x)}} - {\ln {(\mu)}}}\rbrack}^{2}}{{2{\lbrack{\ln {(\sigma)}}\rbrack}}^{2}}}}$for  x > 0; μ > 0  and  σ > 0${mean} = {\exp\left( {\mu + \frac{\sigma^{2}}{2}} \right)}$${{standard}\mspace{14mu} {deviation}} = \sqrt{{\exp \left( {\sigma^{2} + {2\; \mu}} \right)}\left\lbrack {{\exp \left( \sigma^{2} \right)} - 1} \right\rbrack}$${skewness} = {\left\lfloor \sqrt{{\exp \left( \sigma^{2} \right)} - 1} \right\rfloor \left( {2 + {\exp \left( \sigma^{2} \right)}} \right)}$excess  kurtosis = exp (4 σ²) + 2exp (3 σ²) + 3 exp (2 σ²) − 6

Mean (μ) and standard deviation (σ) are the distributional parameters.The input requirements are such that Mean and Standard deviation areboth>0 and can be any positive value. By default, the lognormaldistribution uses the arithmetic mean and standard deviation. Forapplications for which historical data are available, it is moreappropriate to use either the logarithmic mean and standard deviation,or the geometric mean and standard deviation.

Normal Distribution

The normal distribution is the most important distribution inprobability theory because it describes many natural phenomena, such aspeople's IQs or heights. Decision makers can use the normal distributionto describe uncertain variables such as the inflation rate or the futureprice of gasoline.

Conditions

The three conditions underlying the normal distribution are:

-   -   Some value of the uncertain variable is the most likely (the        mean of the distribution).    -   The uncertain variable could as likely be above the mean as it        could be below the mean (symmetrical about the mean).    -   The uncertain variable is more likely to be in the vicinity of        the mean than further away.

The mathematical constructs for the normal distribution are as follows:

${{f(x)} = {\frac{1}{\sqrt{2\; \pi}\sigma}^{- \frac{{({x - \mu})}^{2}}{2\; \sigma^{2}}}\mspace{14mu} {for}\mspace{14mu} {all}\mspace{14mu} {values}\mspace{14mu} {of}\mspace{14mu} x\mspace{14mu} {and}\mspace{14mu} \mu}};$while  σ > mean = μ standard  deviation = σ skewness = 0(this  applies  to  all  inputs  of  mean  and  standard  deviation)excess  kurtosis = 0(this  applies  to  all  inputs  of  mean  and  standard  deviation)

Mean (μ) and standard deviation (σ) are the distributional parameters.The input requirements are such that Standard deviation>0 and can be anypositive value and Mean can be any value.

Pareto Distribution

The Pareto distribution is widely used for the investigation ofdistributions associated with such empirical phenomena as citypopulation sizes, the occurrence of natural resources, the size ofcompanies, personal incomes, stock price fluctuations, and errorclustering in communication circuits.

The mathematical constructs for the pareto are as follows:

${f(x)} = {{\frac{\beta \; L^{\beta}}{x^{({1 + \beta})}}\mspace{14mu} {for}\mspace{14mu} x} > L}$${mean} = \frac{\beta \; L}{\beta - 1}$${{standard}\mspace{14mu} {deviation}} = \sqrt{\frac{\beta \; L^{2}}{\left( {\beta - 1} \right)^{2}\left( {\beta - 2} \right)}}$${skewness} = {\sqrt{\frac{\beta - 2}{\beta}}\left\lbrack \frac{2\left( {\beta + 1} \right)}{\beta - 3} \right\rbrack}$${{excess}\mspace{14mu} {kurtosis}} = \frac{6\left( {\beta^{3} + \beta^{2} - {6\; \beta} - 2} \right)}{{\beta \left( {\beta - 3} \right)}\left( {\beta - 4} \right)}$

Location (L) and shape (β) are the distributional parameters.

There are two standard parameters for the Pareto distribution: locationand shape. The location parameter is the lower bound for the variable.After you select the location parameter, you can estimate the shapeparameter. The shape parameter is a number greater than 0, usuallygreater than 1. The larger the shape parameter, the smaller the varianceand the thicker the right tail of the distribution. The inputrequirements are such that Location>0 and can be any positive valuewhile Shape≧0.05.

Student's t Distribution

The Student's t distribution is the most widely used distribution inhypothesis test. This distribution is used to estimate the mean of anormally distributed population when the sample size is small, and isused to test the statistical significance of the difference between twosample means or confidence intervals for small sample sizes.

The mathematical constructs for the t-distribution are as follows:

${f(t)} = {\frac{\Gamma \left\lbrack {\left( {r + 1} \right)/2} \right\rbrack}{\sqrt{r\; \pi}{\Gamma \left\lbrack {r/2} \right\rbrack}}\left( {1 + {t^{2}/r}} \right)^{{- {({r + 1})}}/2}}$mean = 0 $\begin{pmatrix}{{this}\mspace{14mu} {applies}\mspace{14mu} {to}\mspace{14mu} {all}\mspace{14mu} {degrees}\mspace{14mu} {of}\mspace{14mu} {freedom}\mspace{14mu} r\mspace{14mu} {except}\mspace{14mu} {if}\mspace{14mu} {the}} \\{{distribution}\mspace{14mu} {is}\mspace{14mu} {shifted}\mspace{14mu} {to}\mspace{14mu} {another}\mspace{14mu} {nonzero}\mspace{14mu} {central}\mspace{14mu} {location}}\end{pmatrix}$${{standard}\mspace{14mu} {deviation}} = \sqrt{\frac{r}{r - 2}}$skewness = 0${{excess}\mspace{14mu} {kurtosis}} = {{\frac{6}{r - 4}\mspace{14mu} {for}\mspace{14mu} {all}} > 4}$

Degree of freedom r is the only distributional parameter. Thet-distribution is related to the F-distribution as follows: the squareof a value of t with r degrees of freedom is distributed as F with 1 andr degrees of freedom. The overall shape of the probability densityfunction of the t-distribution also resembles the bell shape of anormally distributed variable with mean 0 and variance 1, except that itis a bit lower and wider or is leptokurtic (fat tails at the ends andpeaked center). As the number of degrees of freedom grows (say, above30), the t-distribution approaches the normal distribution with mean 0and variance 1. The input requirements are such that Degrees offreedom≧1 and must be an integer.

Triangular Distribution

The triangular distribution describes a situation where you know theminimum, maximum, and most likely values to occur. For example, youcould describe the number of cars sold per week when past sales show theminimum, maximum, and usual number of cars sold.

Conditions

The three conditions underlying the triangular distribution are:

-   -   The minimum number of items is fixed.    -   The maximum number of items is fixed.    -   The most likely number of items falls between the minimum and        maximum values, forming a triangular-shaped distribution, which        shows that values near the minimum and maximum are less likely        to occur than those near the most-likely value.

The mathematical constructs for the triangular distribution are asfollows:

$\mspace{20mu} {{f(x)} = \left\{ {{\begin{matrix}\frac{2\left( {x - {Min}} \right)}{\left( {{Max} - {Min}} \right)\left( {{Likely} - \min} \right)} & {{{for}\mspace{14mu} {Min}} < x < {Likely}} \\\frac{2\left( {{Max} - x} \right)}{\left( {{Max} - {Min}} \right)\left( {{Max} - {Likely}} \right)} & {{{for}\mspace{14mu} {Likely}} < x < {Max}}\end{matrix}\mspace{20mu} {mean}} = {{\frac{1}{3}\left( {{Min} + {Likely} + {Max}} \right){standard}{\mspace{11mu} \;}{deviation}} = {{\sqrt{\frac{1}{18}\begin{pmatrix}{{Min}^{2} + {Likely}^{2} + {Max}^{2} -} \\{{MinMax} - {{MinL}{ikely}} - {{MaxL}{ikely}}}\end{pmatrix}}\mspace{20mu} {skewness}} = {{\frac{\begin{matrix}{\sqrt{2}\left( {{Min} + {Max} - {2\; {Likely}}} \right)\left( {{2\; {Min}} - {Max} - {Likely}} \right)} \\\left( {{Min} - {2\; {Max}} + {Likely}} \right)\end{matrix}}{5\begin{pmatrix}{{Min}^{2} + {Max}^{2} + {Likely}^{2} - {MinMax} -} \\{{MinLikely} - {MaxLikely}}\end{pmatrix}^{3/2}}\mspace{20mu} {excess}\mspace{14mu} {kurtosis}} = {- 0.6}}}}} \right.}$

Minimum (Min), most likely (Likely) and maximum (Max) are thedistributional parameters and the input requirements are such thatMin≦Most Likely≦Max and can take any value, Min<Max and can take anyvalue.

Uniform Distribution

With the uniform distribution, all values fall between the minimum andmaximum and occur with equal likelihood.

The three conditions underlying the uniform distribution are:

-   -   The minimum value is fixed.    -   The maximum value is fixed.    -   All values between the minimum and maximum occur with equal        likelihood.

The mathematical constructs for the uniform distribution are as follows:

${f(x)} = \frac{1}{{Max} - {Min}}$for  all  values  such  that  Min < Max${mean} = \frac{{Min} + {Max}}{2}$${{standard}\mspace{14mu} {deviation}} = \sqrt{\frac{\left( {{Max} - {Min}} \right)^{2}}{12}}$skewness = 0 excess  kurtosis = −1.2(this  applies  to  all  inputs  of  Min  and  Max)

Maximum value (Max) and minimum value (Min) are the distributionalparameters. The input requirements are such that Min<Max and can takeany value.

Weibull Distribution (Rayleigh Distribution)

The Weibull distribution describes data resulting from life and fatiguetests. It is commonly used to describe failure time in reliabilitystudies as well as the breaking strengths of materials in reliabilityand quality control tests. Weibull distributions are also used torepresent various physical quantities, such as wind speed. The Weibulldistribution is a family of distributions that can assume the propertiesof several other distributions. For example, depending on the shapeparameter you define, the Weibull distribution can be used to model theexponential and Rayleigh distributions, among others. The Weibulldistribution is very flexible. When the Weibull shape parameter is equalto 1.0, the Weibull distribution is identical to the exponentialdistribution. The Weibull location parameter lets you set up anexponential distribution to start at a location other than 0.0. When theshape parameter is less than 1.0, the Weibull distribution becomes asteeply declining curve. A manufacturer might find this effect useful indescribing part failures during a burn-in period.

The mathematical constructs for the Weibull distribution are as follows:

${f(x)} = {{\frac{\alpha}{\beta}\left\lbrack \frac{x}{\beta} \right\rbrack}^{\alpha - 1}^{- {(\frac{x}{\beta})}^{\alpha}}}$mean = β Γ(1 + α⁻¹)standard  deviation = β²[Γ(1 + 2 α⁻¹) − Γ²(1 + α⁻¹)]${skewness} = \frac{\begin{matrix}{{2\; {\Gamma^{3}\left( {1 + \beta^{- 1}} \right)}} - {3\; {\Gamma \left( {1 + \beta^{- 1}} \right)}}} \\{{\Gamma \left( {1 + {2\; \beta^{- 1}}} \right)} + {\Gamma \left( {1 + {3\; \beta^{- 1}}} \right)}}\end{matrix}}{\left\lbrack {{\Gamma \left( {1 + {2\; \beta^{- 1}}} \right)} - {\Gamma^{2}\left( {1 + \beta^{- 1}} \right)}} \right\rbrack^{3/2}}$${{excess}\mspace{14mu} {kurtosis}} = \frac{\begin{matrix}{{{- 6}\; {\Gamma^{4}\left( {1 + \beta^{- 1}} \right)}} + {12\; {\Gamma^{2}\left( {1 + \beta^{- 1}} \right)}{\Gamma \left( {1 + {2\; \beta^{- 1}}} \right)}} -} \\{{3\; {\Gamma^{2}\left( {1 + {2\; \beta^{- 1}}} \right)}} - {4\; {\Gamma \left( {1 + \beta^{- 1}} \right)}{\Gamma \left( {1 + {3\; \beta^{- 1}}} \right)}} +} \\{\Gamma \left( {1 + {4\; \beta^{- 1}}} \right)}\end{matrix}}{\left\lbrack {{\Gamma \left( {1 + {2\; \beta^{- 1}}} \right)} - {\Gamma^{2}\left( {1 + \beta^{- 1}} \right)}} \right\rbrack^{2}}$

Location (L), shape (α) and scale (β) are the distributional parameters,and Γ is the Gamma function. The input requirements are such thatScale>0 and can be any positive value, Shape≧0.05 andLocation can take on any value.

Multiple Regression Analysis and Econometric Data Analysis

This section demonstrates the mathematical models and computations usedin creating the general regression equations, which take the form ofY=β₀+β₁X₁+β₂X₂+ . . . +β_(n)X_(n)+ε where β₀ is the intercept, β_(i) arethe slope coefficients, and ε is the error term. The Y term is thedependent variable and the X terms are the independent variables, wherethese X variables are also known as the regressors. The dependentvariable is named as such as it depends on the independent variable, forexample, sales revenue depends on the amount of marketing costs expendedon a product's advertising and promotion, making the dependent variablesales and the independent variable marketing costs. An example of abivariate regression where there is only a single Y and a single Xvariable, is seen as simply inserting the best-fitting line through aset of data points in a two-dimensional plane. In other cases, amultivariate regression can be performed, where there are multiple or knumber of independent X variables or regressors where in this case, thebest-fitting line will be within a k+1 dimensional plane.

Fitting a line through a set of data points in a multidimensionalscatter plot may result in numerous possible lines. The best-fittingline is defined as the single unique line that minimizes the totalvertical errors, that is, the sum of the absolute distances between theactual data points (Y_(i)) and the estimated line (Ŷ). To find thebest-fitting unique line that minimizes the errors, a more sophisticatedapproach is applied, using multivariate regression analysis. Regressionanalysis therefore finds the unique best-fitting line by requiring thatthe total errors be minimized, or by calculating

${Min}{\sum\limits_{i = 1}^{n}\; \left( {Y_{i} - {\hat{Y}}_{i}} \right)^{2}}$

Only one unique line will minimize this sum of squared errors as shownin the equation above. The errors (vertical distances between the actualdata and the predicted line) are squared to avoid the negative errorsfrom canceling out the positive errors. Solving this minimizationproblem with respect to the slope and intercept requires calculatingfirst derivatives and setting them equal to zero:

${\frac{}{\beta_{0}}{\sum\limits_{i = 1}^{n}\; \left( {Y_{i} - {\hat{Y}}_{i}} \right)^{2}}} = {{0\mspace{14mu} {and}\mspace{14mu} \frac{}{\beta_{1}}{\sum\limits_{i = 1}^{n}\; \left( {Y_{i} - {\hat{Y}}_{i}} \right)^{2}}} = 0}$

Which yields the simple bivariate regression's set of least squaresequations:

$\beta_{1} = {\frac{\sum\limits_{i = 1}^{n}\; {\left( {X_{i} - \overset{\_}{X}} \right)\left( {Y_{i} - \overset{\_}{Y}} \right)}}{\sum\limits_{i = 1}^{n}\; \left( {X_{i} - \overset{\_}{X}} \right)^{2}} = \frac{{\sum\limits_{i = 1}^{n}\; {X_{i}Y_{i}}} - \frac{\sum\limits_{i = 1}^{n}\; {X_{i}{\sum\limits_{i = 1}^{n}\; Y_{i}}}}{n}}{{\sum\limits_{i = 1}^{n}\; X_{i}^{2}} - \frac{\left( {\sum\limits_{i = 1}^{n}\; X_{i}} \right)^{2}}{n}}}$$\beta_{0} = {\overset{\_}{Y} - {\beta_{1}\overset{\_}{X}}}$

For multivariate regression, the analogy is expanded to account formultiple independent variables, where Y_(i)=β₁+β₂X_(2,i)+β₃X_(3,i)+ε_(i)and the estimated slopes can be calculated by:

${\hat{\beta}}_{2} = \frac{{\sum\; {Y_{i}X_{2,i}{\sum\; X_{3,i}^{2}}}} - {\sum\; {Y_{i}X_{3,i}{\sum\; {X_{2,i}X_{3,i}}}}}}{{\sum\; {X_{2,i}^{2}{\sum\; X_{3,i}^{2}}}} - \left( {\sum\; {X_{2,i}X_{3,i}}} \right)^{2}}$${\hat{\beta}}_{3} = \frac{{\sum\; {Y_{i}X_{3,i}{\sum\; X_{2,i}^{2}}}} - {\sum\; {Y_{i}X_{2,i}{\sum\; {X_{2,i}X_{3,i}}}}}}{{\sum\; {X_{2,i}^{2}{\sum\; X_{3,i}^{2}}}} - \left( {\sum\; {X_{2,i}X_{3,i}}} \right)^{2}}$

This set of results can be summarized using matrix notations:[X′X]⁻¹[X′Y].

In running multivariate regressions, great care must be taken to set upand interpret the results. For instance, a good understanding ofeconometric modeling is required (e.g., identifying regression pitfallssuch as structural breaks, multicollinearity, heteroskedasticity,autocorrelation, specification tests, nonlinearities, and so forth)before a proper model can be constructed. Therefore the presentinvention includes some advanced econometrics approaches that are basedon the principles of multiple regression outlined above.

One approach used is that of an Auto-ARIMA, which is based on thefundamental concepts of ARIMA theory or Autoregressive Integrated MovingAverage models. ARIMA(p,d,q) models are the extension of the AR modelthat uses three components for modeling the serial correlation in thetime series data. The first component is the autoregressive (AR) term.The AR(p) model uses the p lags of the time series in the equation. AnAR(p) model has the form: y_(t)=a_(I) ^(y) _(t-l)+ . . .+a_(p)y_(t-p)+e_(t). The second component is the integration (d) orderterm. Each integration order corresponds to differencing the timeseries. I(1) means differencing the data once. I (d) means differencingthe data d times. The third component is the moving average (MA) term.The MA(q) model uses the q lags of the forecast errors to improve theforecast. An MA(q) model has the form: y_(t)=e_(t)+b_(l)e_(t-l)+ . . .+b_(q)e_(t-q). Finally, an ARMA(p,q) model has the combined form:y_(t)=a_(l)y_(t-l)+ . . . +a_(p)y_(t-p)+e_(t)+b_(l)e_(t-l)+ . . .+b_(q)e_(t-q). Using this ARIMA concept, various combinations of p, d, qintegers are tested in an automated and systematic fashion to determinethe best-fitting model for the user's data.

In order to determine the best fitting model, we apply severalgoodness-of-fit statistics to provide a glimpse into the accuracy andreliability of the estimated regression model. They usually take theform of a t-statistic, F-statistic, R-squared statistic, adjustedR-squared statistic, Durbin-Watson statistic, Akaike Criterion, SchwarzCriterion, and their respective probabilities.

The R-squared (R²), or coefficient of determination, is an errormeasurement that looks at the percent variation of the dependentvariable that can be explained by the variation in the independentvariable for a regression analysis. The coefficient of determination canbe calculated by:

$R^{2} = {{1 - \frac{\sum\limits_{i = 1}^{n}\; \left( {Y_{i} - {\hat{Y}}_{i}} \right)^{2}}{\sum\limits_{i = 1}^{n}\; \left( {Y_{i} - \overset{\_}{Y}} \right)^{2}}} = {1 - \frac{S\; S\; E}{T\; S\; S}}}$

Where the coefficient of determination is one less the ratio of the sumsof squares of the errors (SSE) to the total sums of squares (TSS). Inother words, the ratio of SSE to TSS is the unexplained portion of theanalysis, thus, one less the ratio of SSE to TSS is the explainedportion of the regression analysis.

The estimated regression line is characterized by a series of predictedvalues (Ŷ); the average value of the dependent variable's data points isdenoted Y; and the individual data points are characterized by Y_(i).Therefore, the total sum of squares, that is, the total variation in thedata or the total variation about the average dependent value, is thetotal of the difference between the individual dependent values and itsaverage (the total squared distance of Y_(i)− Y). The explained sum ofsquares, the portion that is captured by the regression analysis, is thetotal of the difference between the regression's predicted value and theaverage dependent variable's data set (seen as the total squareddistance of Ŷ− Y). The difference between the total variation (TSS) andthe explained variation (ESS) is the unexplained sums of squares, alsoknown as the sums of squares of the errors (SSE).

Another related statistic, the adjusted coefficient of determination, orthe adjusted R-squared ( R ²), corrects for the number of independentvariables (k) in a multivariate regression through a degrees of freedomcorrection to provide a more conservative estimate:

${\overset{\_}{R}}^{2} = {{1 - \frac{\sum\limits_{i = 1}^{n}\; {\left( {Y_{i} - {\hat{Y}}_{i}} \right)^{2}/\left( {k - 2} \right)}}{\sum\limits_{i = 1}^{n}\; {\left( {Y_{i} - \overset{\_}{Y}} \right)^{2}/\left( {k - 1} \right)}}} = {1 - \frac{S\; S\; {E/\left( {k - 2} \right)}}{T\; S\; {S/\left( {k - 1} \right)}}}}$

The adjusted R-squared should be used instead of the regular R-squaredin multivariate regressions because every time an independent variableis added into the regression analysis, the R-squared will increase;indicating that the percent variation explained has increased. Thisincrease occurs even when nonsensical regressors are added. The adjustedR-squared takes the added regressors into account and penalizes theregression accordingly, providing a much better estimate of a regressionmodel's goodness-of-fit.

Other goodness-of-fit statistics include the t-statistic and theF-statistic. The former is used to test if each of the estimated slopeand intercept(s) is statistically significant, that is, if it isstatistically significantly different from zero (therefore making surethat the intercept and slope estimates are statistically valid). Thelatter applies the same concepts but simultaneously for the entireregression equation including the intercept and slopes. Using theprevious example, the following illustrates how the t-statistic andF-statistic can be used in a regression analysis.

When running the Autoeconometrics methodology, multiple regressionissues and errors are first tested for. These include items such asheteroskedasticity, multicollinearity, micronumerosity, lags, leads,autocorrelation and others. For instance, several tests exist to testfor the presence of heteroskedasticity. These tests also are applicablefor testing misspecifications and nonlinearities. The simplest approachis to graphically represent each independent variable against thedependent variable as illustrated earlier. Another approach is to applyone of the most widely used model, the White's test, where the test isbased on the null hypothesis of no heteroskedasticity against analternate hypothesis of heteroskedasticity of some unknown general form.The test statistic is computed by an auxiliary or secondary regression,where the squared residuals or errors from the first regression areregressed on all possible (and nonredundant) cross products of theregressors. For example, suppose the following regression is estimated:

Y=β ₀+β₁ X+β ₂ Z+ε _(t)

The test statistic is then based on the auxiliary regression of theerrors (ε):

ε_(t) ²=α₀+α₁ X+α ₂ Z+α ₃ X ²+α₄ Z ²+α₅ XZ+v _(t)

The nR² statistic is the White's test statistic, computed as the numberof observations (n) times the centered R-squared from the testregression. White's test statistic is asymptotically distributed as a χ²with degrees of freedom equal to the number of independent variables(excluding the constant) in the test regression.

The White's test is also a general test for model misspecification,because the null hypothesis underlying the test assumes that the errorsare both homoskedastic and independent of the regressors, and that thelinear specification of the model is correct. Failure of any one ofthese conditions could lead to a significant test statistic. Conversely,a nonsignificant test statistic implies that none of the threeconditions is violated. For instance, the resulting F-statistic is anomitted variable test for the joint significance of all cross products,excluding the constant.

One method to fix heteroskedasticity is to make it homoskedastic byusing a weighted least squares (WLS) approach. For instance, suppose thefollowing is the original regression equation:

Y=β ₀+β₁ X ₁+β₂ X ₂+β₃ X ₃+ε

Further suppose that X₂ is heteroskedastic. Then transform the data usedin the regression into:

$Y = {\frac{\beta_{0}}{X_{2}} + {\beta_{1}\frac{X_{1}}{X_{2}}} + \beta_{2} + {\beta_{3}\frac{X_{3}}{X_{2}}} + \frac{ɛ}{X_{2}}}$

The model can be redefined as the following WLS regression:

Y _(WLS)=β₀ ^(WLS)+β₁ ^(WLS) X ₁+β₂ ^(WLS) X ₂+β₃ ^(WLS) X ₃ +v

Alternatively, the Park's test can be applied to test forheteroskedasticity and to fix it. The Park's test model is based on theoriginal regression equation, uses its errors, and creates an auxiliaryregression that takes the form of:

ln e _(i) ²=β₁+β₂ ln X _(k,i)

Suppose β₂ is found to be statistically significant based on a t-test,then heteroskedasticity is found to be present in the variable X_(k,i).The remedy therefore is to use the following regression specification:

$\frac{Y}{\sqrt{X_{k}^{\beta_{2}}}} = {\frac{\beta_{1}}{\sqrt{X_{k}^{\beta_{2}}}} + \frac{\beta_{2}X_{2}}{\sqrt{X_{k}^{\beta_{2}}}} + \frac{\beta_{3}X_{3}}{\sqrt{X_{k}^{\beta_{2}}}} + {ɛ.}}$

Multicollinearity exists when there is a linear relationship between theindependent variables. When this occurs, the regression equation cannotbe estimated at all. In near collinearity situations, the estimatedregression equation will be biased and provide inaccurate results. Thissituation is especially true when a step-wise regression approach isused, where the statistically significant independent variables will bethrown out of the regression mix earlier than expected, resulting in aregression equation that is neither efficient nor accurate.

As an example, suppose the following multiple regression analysisexists, where

Y _(i)=β₁+β₂ X _(2,i)+β₃ X _(3,i)+ε_(i)

The estimated slopes can be calculated through

${\hat{\beta}}_{2} = \frac{{\sum\; {Y_{i}X_{2,i}{\sum\; X_{3,i}^{2}}}} - {\sum\; {Y_{i}X_{3,i}{\sum\; {X_{2,i}X_{3,i}}}}}}{{\sum\; {X_{2,i}^{2}{\sum\; X_{3,i}^{2}}}} - \left( {\sum\; {X_{2,i}X_{3,i}}} \right)^{2}}$${\hat{\beta}}_{3} = \frac{{\sum\; {Y_{i}X_{3,i}{\sum\; X_{2,i}^{2}}}} - {\sum\; {Y_{i}X_{2,i}{\sum\; {X_{2,i}X_{3,i}}}}}}{{\sum\; {X_{2,i}^{2}{\sum\; X_{3,i}^{2}}}} - \left( {\sum\; {X_{2,i}X_{3,i}}} \right)^{2}}$

Now suppose that there is perfect multicollinearity, that is, thereexists a perfect linear relationship between X₂ and X₃, such thatX_(3,i)=λX_(2,i) for all positive values of λ. Substituting this linearrelationship into the slope calculations for β₂, the result isindeterminate. In other words, we have

${\hat{\beta}}_{2} = {\frac{{\sum\; {Y_{i}X_{2,i}{\sum\; {\lambda^{2}X_{2,i}^{2}}}}} - {\sum\; {Y_{i}\lambda \; X_{2,i}{\sum\; {\lambda \; X_{2,i}^{2}}}}}}{{\sum\; {X_{2,i}^{2}{\sum\; {\lambda^{2}X_{2,i}^{2}}}}} - \left( {\sum\; {\lambda \; X_{2,i}^{2}}} \right)^{2}} = \frac{0}{0}}$

The same calculation and results apply to β₃, which means that themultiple regression analysis breaks down and cannot be estimated given aperfect collinearity condition.

One quick test of the presence of multicollinearity in a multipleregression equation is that the R-squared value is relatively high whilethe t-statistics are relatively low. Another quick test is to create acorrelation matrix between the independent variables. A high crosscorrelation indicates a potential for multicollinearity. The rule ofthumb is that a correlation with an absolute value greater than 0.75 isindicative of severe multicollinearity.

Another test for multicollinearity is the use of the variance inflationfactor (VIF), obtained by regressing each independent variable to allthe other independent variables, obtaining the R-squared value andcalculating the VIF of that variable by estimating:

${V\; I\; F_{i}} = \frac{1}{\left( {1 - R_{i}^{2}} \right)}$

A high VIF value indicates a high R-squared near unity. As a rule ofthumb, a VIF value greater than 10 is usually indicative of destructivemulticollinearity. The Autoeconometrics method computes formulticollinearity and corrects the data before running the nextiteration when enumerating through the entire set of possiblecombinations and permutations of models.

One very simple approach to test for autocorrelation is to graph thetime series of a regression equation's residuals. If these residualsexhibit some cyclicality, then autocorrelation exists. Another morerobust approach to detect autocorrelation is the use of theDurbin-Watson statistic, which estimates the potential for a first-orderautocorrelation. The Durbin-Watson test also identifies modelmisspecification. That is, if a particular time-series variable iscorrelated to itself one period prior. Many time-series data tend to beautocorrelated to their historical occurrences. This relationship can bedue to multiple reasons, including the variables' spatial relationships(similar time and space), prolonged economic shocks and events,psychological inertia, smoothing, seasonal adjustments of the data, andso forth.

The Durbin-Watson statistic is estimated by the sum of the squares ofthe regression errors for one period prior, to the sum of the currentperiod's errors:

${D\; W} = \frac{\sum\; \left( {ɛ_{t} - ɛ_{t - 1}} \right)^{2}}{\sum\; ɛ_{t}^{2}}$

There is a Durbin-Watson critical statistic table at the end of the bookthat provides a guide as to whether a statistic implies anyautocorrelation.

Another test for autocorrelation is the Breusch-Godfrey test, where fora regression function in the form of:

Y=f(X ₁ ,X ₂ . . . , X _(k))

Estimate this regression equation and obtain its errors ε_(t). Then, runthe secondary regression function in the form of:

Y=f(X ₁ ,X ₂ . . . , X _(k),ε_(t-1),ε_(t-2),ε_(t-p))

Obtain the R-squared value and test it against a null hypothesis of noautocorrelation versus an alternate hypothesis of autocorrelation, wherethe test statistic follows a Chi-Square distribution of p degrees offreedom:

R ²(n−p)˜Ω_(df=p) ²

Fixing autocorrelation requires the application of advanced econometricmodels including the applications of ARIMA (as described above) or ECM(Error Correction Models). However, one simple fix is to take the lagsof the dependent variable for the appropriate periods, add them into theregression function, and test for their significance, for instance:

Y _(t) =f(Y _(t-1) ,Y _(t-2) , . . . , Y _(t-p) ,X ₁ ,X ₂ , . . . , X_(k))

In interpreting the results of an Autoeconometrics model, most of thespecifications are identical to the multivariate regression analysis.However, there are several additional sets of results specific to theeconometric analysis. The first is the addition of Akaike InformationCriterion (AIC) and Schwarz Criterion (SC), which are often used inARIMA model selection and identification. That is, AIC and SC are usedto determine if a particular model with a specific set of p, d, and qparameters is a good statistical fit. SC imposes a greater penalty foradditional coefficients than the AIC but generally, the model with thelowest AIC and SC values should be chosen. Finally, an additional set ofresults called the autocorrelation (AC) and partial autocorrelation(PAC) statistics are provided in the ARIMA report.

For instance, if autocorrelation AC(1) is nonzero, it means that theseries is first order serially correlated. If AC dies off more or lessgeometrically with increasing lags, it implies that the series follows alow-order autoregressive process. If AC drops to zero after a smallnumber of lags, it implies that the series follows a low-ordermoving-average process. In contrast, PAC measures the correlation ofvalues that are k periods apart after removing the correlation from theintervening lags. If the pattern of autocorrelation can be captured byan autoregression of order less than k, then the partial autocorrelationat lag k will be close to zero. The Ljung-Box Q-statistics and theirp-values at lag k are also provided, where the null hypothesis beingtested is such that there is no autocorrelation up to order k. Thedotted lines in the plots of the autocorrelations are the approximatetwo standard error bounds. If the autocorrelation is within thesebounds, it is not significantly different from zero at approximately the5% significance level. Finding the right ARIMA model takes practice andexperience. These AC, PAC, SC, and AIC are highly useful diagnostictools to help identify the correct model specification. Finally, theARIMA parameter results are obtained using sophisticated optimizationand iterative algorithms, which means that although the functional formslook like those of a multivariate regression, they are not the same.ARIMA is a much more computationally intensive and advanced econometricapproach.

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